If `|Z-2|=2|Z-1|`, then the value of `(Re(Z))/(|Z|^(2))` is (where Z is a complex number and `Re(Z)` represents the real part of Z)

Answer

To solve the problem given by the equation \( |Z - 2| = 2|Z - 1| \), where \( Z \) is a complex number, we will follow these steps:

### Step 1: Express \( Z \) in terms of its real and imaginary parts.
Let \( Z = a + ib \), where \( a \) is the real part and \( b \) is the imaginary part of \( Z \).

### Step 2: Rewrite the equation using the definition of modulus.
The left-hand side becomes:
\[ |Z - 2| = |(a - 2) + ib| = \sqrt{(a - 2)^2 + b^2} \]
The right-hand side becomes:
\[ 2|Z - 1| = 2|(a - 1) + ib| = 2\sqrt{(a - 1)^2 + b^2} \]
Thus, we can rewrite the equation as:
\[ \sqrt{(a - 2)^2 + b^2} = 2\sqrt{(a - 1)^2 + b^2} \]

### Step 3: Square both sides to eliminate the square roots.
Squaring both sides gives:
\[ (a - 2)^2 + b^2 = 4((a - 1)^2 + b^2) \]

### Step 4: Expand both sides.
Expanding the left-hand side:
\[ (a - 2)^2 + b^2 = a^2 - 4a + 4 + b^2 \]
Expanding the right-hand side:
\[ 4((a - 1)^2 + b^2) = 4(a^2 - 2a + 1 + b^2) = 4a^2 - 8a + 4 + 4b^2 \]

### Step 5: Set the expanded forms equal to each other.
Now we have:
\[ a^2 - 4a + 4 + b^2 = 4a^2 - 8a + 4 + 4b^2 \]

### Step 6: Rearrange the equation.
Rearranging gives:
\[ a^2 - 4a + 4 + b^2 - 4a^2 + 8a - 4 - 4b^2 = 0 \]
This simplifies to:
\[ -3a^2 + 4a - 3b^2 = 0 \]

### Step 7: Factor out common terms.
Rearranging gives:
\[ 3a^2 + 3b^2 = 4a \]
Dividing through by 3:
\[ a^2 + b^2 = \frac{4}{3}a \]

### Step 8: Express \( \frac{Re(Z)}{|Z|^2} \).
We know that \( |Z|^2 = a^2 + b^2 \). From the previous step, we have:
\[ |Z|^2 = \frac{4}{3}a \]
Thus:
\[ \frac{Re(Z)}{|Z|^2} = \frac{a}{|Z|^2} = \frac{a}{\frac{4}{3}a} = \frac{3}{4} \]

### Final Answer:
The value of \( \frac{Re(Z)}{|Z|^2} \) is \( \frac{3}{4} \).